Crossover from gas-like to liquid-like molecular diffusion in a simple supercritical fluid

According to textbooks, no physical observable can be discerned allowing to distinguish a liquid from a gas beyond the critical point. Yet, several proposals have been put forward challenging this view and various transition boundaries between a gas-like and a liquid-like behaviour, including the so-called Widom and Frenkel lines, and percolation line, have been suggested to delineate the supercritical state space. Here we report observation of a crossover from gas-like (Gaussian) to liquid-like (Lorentzian) self-dynamic structure factor by incoherent quasi-elastic neutron scattering measurements on supercritical fluid methane as a function of pressure, along the 200 K isotherm. The molecular self-diffusion coefficient was derived from the best Gaussian (at low pressures) or Lorentzian (at high pressures) fits to the neutron spectra. The Gaussian-to-Lorentzian crossover is progressive and takes place at about the Widom line intercept (59 bar). At considerably higher pressures, a liquid-like jump diffusion mechanism properly describes the supercritical fluid on both sides of the Frenkel line. The present observation of a gas-like to liquid-like crossover in the self dynamics of a simple supercritical fluid confirms emerging views on the unexpectedly complex physics of the supercritical state, and could have planet-wide implications and possible industrial applications in green chemistry.


Pressure (bar)
Supplementary Figure 2: Slope parameter K appearing in Eq. 4 of the main text.Pressure dependence of the slope of the linear fits of the Gaussian half widths shown in Fig. 3a of the main text.67.5 bar is also included.The arrow indicates the value calculated from the theoretical expression for ballistic diffusion (Eq. 1 of the main text): K=(2ln(2)k B T /m) 1/2 .Supplementary Table 1.Pressure P (measured within a 1 bar uncertainty) and density ρ deduced from the literature equation of state as well as slope K, self-diffusion coefficient D, and residence time τ with their fractional uncertainties (one standard deviation) as well as literature shear viscosity η.All columns are for methane at 200 K. P (bar) ρ (g cm -3 ) hK (meV Å) δ(hK)(%) D (10 -9 m 2 s -1 ) δD(%) τ (ps) δτ (%) η (10 - 6  Supplementary Note 1: Locating the Gaussian-to-Lorentzian crossover upon pressure increase The Gaussian-to-Lorentzian crossover cannot be easily located in pressure based on the comparison of the chi-square values of the fits of the spectra alone, which are Q dependent and also partially dependent on other fitting parameters.Some other considerations can be made to locate the crossover, as reported here below, and the chi-square will be discussed in the context of the Bayesian analysis of the spectra (Supplementary Note 2).
First, we observe that for a given wavevector transfer value, the normalized experimental spectra cross as a consequence of the pressure-induced Gaussian-to-Lorentzian change in the quasi-elastic lineshape.This is shown for example for Q=1 Å-1 in Supplementary Fig. 1, where the crossing happens at about -3.5 meV when comparing the 46 bar and 51.5 bar spectra as well as the 51.5 bar and the 67.5 bar spectra.This would then locate the Gaussian-to-Lorentzian crossover in between 46 and 67.5 bar.
Second, even though the Gaussian and Lorentzian fits to the spectra are of comparable quality at 40-60 bar (see Fig. 2 of the main text), for the Lorentzian fits, the flat background had to be imposed to zero (it would otherwise converge to negative values), as shown in Supplementary Fig.

13.
Third, being Q-independent, the likeliness of the diffusion coefficients obtained from the Gaussian and Lorentzian fits is a rather strong indication of the crossover.As briefly mentioned in the main text in relation to Fig. 4, the largest relative deviations between our experimental D and the literature 1 molecular dynamics hard-spheres prediction for D are for the intermediate pressure range (40-80 bar).This is quantified in Supplementary Fig. 3, which indeed shows the relative deviations as a function of pressure and includes the Lorentzian estimations at 46 and 51.5 bar, and the Gaussian estimation at 67.5 bar.Given that the hard spheres results are very precise and well established, this approach provides a robust indication in locating the crossover at about 60 bar (see Supplementary Fig. 3).
Finally, on a purely theoretical ground, particles in a system are expected to behave as free when probed over a spatial range much shorter than the mean free path this should be only taken as a rough indication, we notice that 51.5 bar is our highest investigated pressure for which we employed the Gaussian fits and also the highest investigated pressure for which l 0 Q >1 over the entire Q range of the present experiment, i.e. 0.4-1.5 Å-1 (see plots in Supplementary Fig. 4).At 65 bar, l 0 Q is slightly smaller than unity for our smallest Q value.At our highest pressure (2450 bar), l 0 Q is smaller than 1 over most of the investigated Q range. !" . 0  3: Relative deviation of D from the hard sphere prediction.Relative difference of our experimental diffusion coefficient D from the hard sphere prediction at the same pressure (both reported in Fig. 4 of the main text), plotted as a function of pressure.From 46 to 67.5 bar, results from both models are reported.The two lines (guides to the eye) cross at about 60 bar.inference, we determined also the chi-square for each of the fits performed.We could thus obtain an independent evidence about the progressive change from a Gaussian-like to a Lorentzian-like nature for the QENS spectra of methane as pressure increases.
The main outcomes of this analysis are: 1.At low pressures (P <40 bar), the Gaussian model is always preferable to the Lorentzian one for Q values lower than 1.1 Å-1 , while they are equivalent for higher Q values.An example for the lowest pressure value (8.5 bar) is reported below in Supplementary Figs. 5 and 6.
2. In the intermediate P range 40-60 bar, the Gaussian model is preferable to the Lorentzian one for Q values lower than 1 Å-1 (i.e. the region where the diffusion coefficient is evaluated), while there is a region between 1 and 1.3 Å-1 where Lorentzian seems preferable, and above they are comparable.As an example, the results of the Bayesian analysis for 46 bar are reported below in Supplementary Figs. 7 and 8.
3. At high pressures (P >67 bar), the Lorentzian fit is preferable for each measured Q value.

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Supplementary Figure 12: Flat background parameter B 0 of the Gaussian fits.B 0 parameter of the Gaussian fits as a function of Q. Freely refined values as red crosses and imposed values as green dots. Supplementary Figure 15: Wavevector transfer dependence of I Gauss 0 and I Lor 0 .Fitted Gaussian (panel a) and Lorentzian (panel b) intensities as a function of Q at all investigated pressures.The Gaussian intensity is lower at 46 bar compared to other comparable pressures because the 46 bar data were taken with the aluminium spacer in the high-pressure cell.The Lorentzian intensity is higher at 67.5 and 75.5 bar compared to the other pressures because those data were taken without aluminium spacer.The best fits to the data at each pressure (Eqs.9 and 10 of the main text) are shown as solid lines and provide the mean displacement u 2 1/2 of the Debye-Waller factor, which is reported in Supplementary Fig. 16.Given the large systematic deviation shown by I Lor 0 , the two lowest Q points were excluded in all the fits.
18'?;::'@123' '>:18'682:;>AB:9<B1;'5672189: !tion from which either the mean value or the statistical mode can be considered, as appropriate, to determine the description of the lineshape most probabilistically justified by the data.For a direct comparison with the presented work, after fitting each dataset with the Bayesian algorithm either with a Gaussian profile or with a Lorenztian one, albeit unorthodox in the context of Bayesian 2301;)@AB);.=.C)01DE)F&G) )96=10H<8.=1;)@AB);.=.C)01DE)F&G Supplementary Figure 11: Test of the Stokes-Einstein-Sutherland relation.Pressure dependence of the product between the self-diffusion coefficient D at 200 K (from this work and from ref. 6) and the viscosity η at the same temperature from the equation of state-like viscosity model of ref. 7. The error bars were obtained by propagating the errors in D only.The Stokes-Einstein-Sutherland relation predicts a constant product Dη along isotherms.

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Supplementary Figure13: Flat background parameter B 0 of the Lorentzian fits.B 0 parameter of the Lorentzian fits as a function of Q. Freely refined values as red crosses and imposed values as green dots.